Mechanism Design

withCollusion and Correlation�

Jean-Jacques La�ont��

David Martimort���

Revised Version: December 98

ABSTRACT

In a public good environment with positively correlated types, we characterize optimal

mechanisms when agents have private information and can enter collusive agreements.

First, we prove a weak-collusion-proof principle according to which there is no restriction

for the principal in o�ering weak-collusion-proof mechanisms. Second, with this principle,

we characterize the set of allocations which satisfy individual and coalitional incentive

constraints. The optimal weakly collusion-proof mechanism calls for distortions away

from �rst-best e�ciency obtained without collusion. Allowing collusion restores continuity

between the correlated and the uncorrelated environments. When the correlation becomes

almost perfect, �rst-best e�ciency is approached. Finally, the optimal collusion-proof

mechanism is strongly rati�able.

Keywords : Mechanism Design, Collusion, Correlation.

� We thank Jeon Doh-Shin, Antoine Faure-Grimaud, Oliver Hart, Eric Maskin, Patrick

Rey, Mike Riordan, Tomas Sjostrom, Yossef Spiegel, Dimitri Vayanos, Robert Wilson,

seminar participants at Humbolt University-Berlin, Bocconi University, University of

Maryland, Paris-Séminaire Roy, University Pompeu Fabra, University of California at

Berkeley, University College London, Stanford Business School, University of Alabama,

three referees and Andrew Postlewaite for helpful comments. Financial support from

INRA's AIP Contracts is gratefully acknowledged by David Martimort.�� Université des Sciences Sociales, GREMAQ and IDEI, Toulouse.��� Université de Pau et des Pays de l'Adour, GREMAQ and IDEI, Toulouse.

1

1 Introduction

The provision of public goods under informational constraints is one of the leading text-

book examples of public economics. How should a society made of several agents with

heterogeneous tastes for a public good design an incentive mechanism to induce truthful

revelation of the agents' valuations? Does e�ciency con�ict with incentives? How can

this con�ict, if any, be solved? Finally, what is the distribution of informational rents

induced by asymmetric information?

One striking feature of most previous works on these issues is that they deal mainly

with the case where agents are unable to form coalitions to collectively manipulate the

decision rule.1 By focusing on the role of individual incentive constraints, an important

dimension of resource allocation in society has been neglected: the formation of groups.

It is particularly troublesome when these coalitions can signi�cantly reduce the e�ciency

of the optimal mechanism designed in the absence of coalition incentive constraints. As

Olson (1965, p. 1) has forcefully emphasized �groups of individuals with common interests

usually attempt to further those common interests [� � �] and are expected to act on behalf

of their common interests much as single individuals are often expected to act on behalf

of their personal interest." Following this argument, the standard theoretical framework

must be amended to allow also for the formation of groups promoting their own collective

goals instead of that of society as a whole. The present paper o�ers a framework in which

the consequences of collusion under asymmetric information on both allocative e�ciency

and the distribution of rents in society can be assessed.

We model a con�ict between a social welfare maximizer and a group of agents who

bene�t from the public good but who do not care about its budgetary cost. Contrary

to the standard assumption in mechanism design, the planner has not a perfect control

of the communication technology so that he cannot prevent the agents from colluding.2

Collusion between these agents is modeled in reduced form as in La�ont and Martimort

(1997). A third-party proposes to the colluding agents a side-mechanism to collectively

manipulate their sending of messages to the government. As suggested above, this third-

party does not internalize the social cost of the project but maximizes only the sum

of the agents' utilities. Lastly, no technology for a credible disclosure of information is

available to the colluding partners. The mere forming of a coalition does not change

1Groves (1973) and Green and La�ont (1977) showed that e�ciency could be achieved under asym-metric information on the agents' valuations with dominant strategy if budget balance is not a concern.Arrow (1979) and Aspremont and Gérard-Varet (1979) showed that budget balance could be achievedunder Bayesian implementation. La�ont and Maskin (1982) and Mailath and Postlewaite (1990) showedthat adding the possibility for the agents to veto the mechanism introduces a real con�ict between ef-�ciency and incentives leading to the underprovision of the public good. Ledyard and Palfrey (1996)show that this con�ict also arises in the absence of participation constraints when incentives con�ict withredistribution concerns.

2See Palfrey (1992) for a discussion of this assumption and some of its implications in the case ofBayesian implementation.

2

informational asymmetries between the agents. Coalition formation takes place under

asymmetric information.

When agents do not collude, their Bayesian-Nash behavior does not put any constraint

on the set of interim individually rational and incentive compatible allocations in a cor-

related information environment. Indeed, as shown by Crémer and McLean (1988) in the

case of auction mechanisms, the existence of even a small amount of correlation between

the agents' valuations for the public good is enough to allow the principal to elicit this

�almost common" information. This result is in sharp contrast with the case of uncorre-

lated information since then e�ciency does con�ict with incentives. Hence, the optimal

levels of public good exhibit discontinuities when the degree of correlation goes to zero.

This costless extraction of the agents' surplus by the principal suggests also that

they are likely to form an active coalition in such a correlated information environment.

Nevertheless, a weak collusion-proofness principle holds in this context: any equilibrium of

the overall game of contract o�er cum coalition formation achieves an outcome which can

be replicated with a weakly collusion-proof grand-mechanism, i.e., a grand-mechanism

such that the null side-contract is a continuation equilibrium of the game of coalition

formation. Since one agent's acceptance of the side-contract depends on the status quo

utility level that he gets from playing non-cooperatively the grand-mechanism o�ered

by the principal, the issue of learning from disagreement arises. Weakly collusion-proof

mechanisms are such that the null side-mechanism is a continuation equilibrium of the

game of coalition formation sustained with passive beliefs.3

The weak collusion-proofness principle provides a tractable description of the set of

perfect Bayesian equilibria of the overall game of contract o�er cum coalition formation.

After having described this set, the principal's welfare is optimized subject to participa-

tion, individual and coalition incentive constraints. Generally, the e�cient levels of public

good can no longer be costlessly implemented even in a correlated environment. Taking

into account coalition incentive constraints, there exists now a trade-o� between e�ciency

and rent extraction. Distortions in the quantities of public good which are produced in

the di�erent states of nature reduce the cost of the binding coalition incentive constraints.

Depending on the degree of correlation, collusion-proofness constraints take quite dif-

ferent forms. For weak positive correlation, collusion-proofness constraints are similar to

those that would arise under symmetric information within the coalition. When the degree

of correlation diminishes, coalition incentive constraints are then less and less binding and

the principal prevents more easily collusion. In the limit of uncorrelated information, the

principal costlessly obtains collusion-proofness and the contractual outcome is the same

as if agents had not been colluding. By adding coalition incentive constraints, one moves

then continuously from the outcome with a strictly positive correlation to the outcome

3Rubinstein (1985) coined this expression for games with asymmetric information in which equilibriumbehavior is sustained with prior beliefs out of the equilibrium path.

3

with no correlation.4

For strong correlation, collusion-proofness constraints under asymmetric information

are instead quite di�erent from those obtained when agents can credibly disclose their in-

formation. When the correlation becomes almost perfect, there is only a small probability

that agents have di�erent valuations for the public good. The principal can shut-down

production in this state of nature at almost no social cost. This breaks the coalition

agreement and almost achieves the �rst-best level of expected welfare.

Since it is implemented with Bayesian strategies, the optimal weakly collusion-proof

contract is sensitive to the exact beliefs that the agents have at the time of playing this

mechanism. Because we focus on the case where valuations for the public good may take

only two values, the equilibrium correspondence of this optimal mechanism as posterior

beliefs change can be fully described. This step of the analysis allows us to discuss

the robustness of the optimal mechanism to a preplay communication stage in which

agents may veto or ratify the truthful play of this mechanism and thereby signal some

information to each other. It is then possible to show that the optimal weakly collusion-

proof mechanism is strongly rati�able in the sense of Cramton and Palfrey (1995).

Collusion in public good mechanisms has been �rst analyzed by Green and La�ont

(1979) who prove that the Groves mechanisms are not robust to coalitions when agents

share freely their information. Still with dominant strategy mechanisms but with a con-

tinuum of types, La�ont and Maskin (1980) show then that only pooling decision rules

can be implemented. Crémer (1996) takes into account asymmetric information within

coalitions and shows that the Groves mechanisms are not robust to size-two coalitions

but not beyond. Similarly, there exist some relatively negative results in the case of Nash

implementation when agents can form coalitions (Maskin (1979)). The Nash environment

can be seen as an extreme case of perfect correlation between the agents' types. Our

focus on Bayesian implementation with two types brings more positive results.5 Under

asymmetric information within the coalition, the principal can implement a much larger

set of allocations in these strongly correlated environments. More generally, our approach

provides a complete description of the set of implementable allocations under collusion.

This paper extends La�ont and Martimort (1997) by stressing the role of correlated

information between the agents as a determinant of the strength of their coalition and by

making no restriction on the set of available mechanisms. In this previous work, we re-

stricted the analysis to the case of anonymous mechanisms and uncorrelated information.6

With no correlation, there always exists a costless weakly collusion-proof implementation

4Interestingly, this result requires neither risk-aversion of the agents' utility functions nor limitedliability constraints on transfers (Robert (1991)).

5See also La�ont and Maskin (1979).6Beside the di�erences in the informational structures and the set of available mechanisms, this pre-

vious paper was dealing with a model of regulation. This latter di�erence is without consequence on theresults.

4

of the second-best non-cooperative outcome if the principal can o�er non-anonymous

Bayesian mechanisms. However, collusion still matters when types are correlated even

without any exogenous restriction on the set of mechanisms. Moreover, working in a

correlated environment, we obtain a full characterization of all transfers in the optimal

weakly collusion-proof mechanism. This characterization allows to compute all the ex

post rents precisely and to characterize the outcome of the grand-mechanism when beliefs

di�er from passive ones. This is an important step towards checking the strong rati�ability

of the mechanism.

Section 2 presents the model. Section 3 discusses the optimal mechanism when agents

do not collude. Section 4 derives the weak collusion-proofness principle and characterizes

weakly collusion-proof allocations. Section 5 describes the optimal weakly collusion-proof

mechanism. Section 6 discusses strong rati�ability. Section 7 concludes.

2 The Model

2.1 Technology, Preferences and Information

We consider the provision of public good in a partial equilibrium model. A positive

amount x of public good can be produced at cost c(�) with c0(�) > 0 and c00(�) > 0. There

are two agents in the economy denoted by Ai; i 2 f1; 2g.7 Each of them derives utility

Ui = �ix � ti from consuming an amount x of public good and paying tax ti. Agent Ai

agrees to participate in the public good mechanism when his participation constraint is

satis�ed.

The agents' valuations for the public good, �i; i 2 f1; 2g, are drawn from a common

knowledge joint distribution on �2 where � = f�; ��g is the common support of �1 and

�2 (�� = �� � �). We refer to the probabilities p(�i; �j) of each state (�i; �j), for (i; j) 2

f1; 2g2; as the common knowledge prior beliefs. To make notation simpler, we also write:

p(��; ��) = p11; p(�; ��) = p(��; �) = p12; p(�; �) = p22; where the equality p(�; ��) = p(��; �)

is derived from the symmetry between the two agents. Finally, to capture the congruence

of the agents' interests, �1 and �2 are positively correlated and p12p11

� p22p12: We denote by

� = p11p22 � p212 the degree of positive correlation (� = 0 for independent types). For

simplifying technicalities, we also assume that p12 � p11.8 The conditional beliefs of agent

Ai on Aj's type (j 6= i) induced by the joint distribution above are the same for both

agents and, slightly abusing notations, are denoted by p.

The government, or principal P , only knows the distribution of the agents' valuations

7Restricting to two agents avoids to consider the formation of subcoalitions and signi�cantly simpli�esnotations. However, our methodology could be extended to more than two agents at the cost of anincrease in complexity.

8This assumption ensures that the optimal collusion-proof mechanism under asymmetric informationnever entails bunching in the case of small correlation. It simpli�es signi�cantly the exposition.

5

for the public good but is uninformed on the exact realizations of these shocks at the

time of choosing the public good mechanism. His objective is to maximize the sum of the

agents' utilities knowing that the de�cit for the production of the public good must be

covered by distortionary taxation raised elsewhere in the economy. Formally, the social

welfare function is written as SW =P2

i=1 Ui � (1 + �)�c(x) �

P2i=1 ti

�; where � is the

exogenous cost of public funds.9

2.2 Mechanisms

The principal proposes a grand-mechanismG to the agents. G maps any pair of messages

(m1;m2) belonging to the product message space M1 �M2 = M (where Mi denotes the

message space used by agent Ai) into a triplet fx; t1; t2g. x denotes the amount of public

good produced (x 2 X = IR+) and ti (i 2 f1; 2g) is the tax paid by agent Ai to the

principal. We denote by G = fx(�); t1(�); t2(�)g this grand-mechanism.10

To make notation simpler in the case of direct mechanisms (M = �� �), we denote

by �x; x and x respectively the levels of public good when both agents claim ��, when their

claims di�er (��; �) and when they both claim �. We also denote by tkl for k; l 2 f1; 2g

the tax paid by an agent whose type is �i = � + (2 � k)�� when the other agent's type

is �j = � + (2 � l)��. Because of symmetry between the agents, the corresponding taxes

are independent of the agents' identity.11

2.3 Coalition Formation

An uninformed third-party, T , proposes a side-mechanism S = f�(�); yi(�)i2f1;2gg to the

agents to induce their collusive behavior.

� �(�) is a collective manipulation of the messages sent to the principal.

� fyi(�)gi2f1;2g is a pair of side-transfers. The third-party is not a source of money and

therefore the coalition's budget is balanced:P2

i=1 yi(�1; �2) = 0 for all (�1; �2) 2 �2.

From the revelation principle, there is no loss of generality in assuming that S is a

9The model is formally equivalent to the La�ont and Tirole (1986) partial equilibrium model ofregulation. It could be possible to build a general equilibrium model endogenizing the value of �. Thiscould be done by introducing a third uninformed agent, say A3, in the analysis. By imposing that thesum of the contributions made by this agent and the group A1�A2 covers exactly the cost of the publicgood as in Aspremont and Gérard-Varet (1978), we would be able to endogenize the value of the budgetconstraint's multiplier. Moreover, because of complete information on A3's valuation for the public good,this agent could be forced to always pay his valuation for the public good. One would then be interestedby the coalition between A1 and A2 against A3.

10Note that we do not restrict a priori the set of mechanisms available to the set of direct mechanisms.Other message spaces than the product of the agents' type spaces �2 can be used by the principal.

11The symmetry of the grand-mechanism is without loss of generality under a non-cooperative behavioras we will see below. We also show in the Appendix that it is without loss of generality in the case ofcollusion for a small correlation. The amount of public good does not need to depend on the identity ofwho has a high valuation for the public good when claims di�er.

6

direct mechanism.12 Therefore �(�) and yk(�) (k 2 f1; 2g) map �2 respectively into the

set of measures on M and the set of balanced side-transfers.

Lastly, T is benevolent and maximizes the sum U1 + U2 of the two colluding agents'

utilities obtained by playing the composition of the grand- and the side-mechanism.13

2.4 Timing of the Game

The timing of the overall game of contract o�er cum coalition formation is as follows (see

also Figure 1 for the game tree):

1. Agents learn their respective valuations for the public good.

2. P proposes a grand-mechanism G. If an agent vetoes the grand-mechanism, all

agents get their reservation utility normalized exogenously at zero.

3. The third-party proposes a side-mechanism S to the agents and a non-cooperative

continuation play of G if anyone refuses this side-contract. If both agents accept

S, agents report their types to the third-party who recommends reports into the

grand-mechanism and who commits to enforce the corresponding side-transfers.

4. Reports are sent into the grand-mechanism. The decision on the size of the public

good is made and taxes are paid by the agents. Side-transfers, if any, are imple-

mented.

The third party's o�er of a side-mechanismS on top of the grand-mechanismG induces

a two stage game �(G;S): In the �rst rati�cation stage, agents simultaneously accept or

refuse the side-mechanism and may thereby signal their types to each other. In the second

communication stage, agents send messages either directly to the principal if at least one

of them has refused the side-mechanism or to the third-party if both have accepted. The

third-party recommends then a collective manipulation of the messages to be sent to the

principal. We denote by E(G;S) the set of perfect Bayesian equilibria of �(G;S).

We are interested in �nding the optimal mechanismG, knowing that the continuation

game of coalition formation consists �rst of a side-mechanism S optimally chosen by the

third-party and second of a rati�cation-communication game �(G;S).

12For any grand-mechanism o�ered by the principal, one can restrict the third-party to use directside-mechanisms at the �nal stage of the game of contract o�er cum coalition formation.

13Using this third-party as a side-contract mechanism designer avoids the di�cult issue of informa-tional leakages through contract o�ers. It eliminates also the problem of �nding an extensive form fordescribing the collusive game between the agents. This third-party paradigm can be seen as a black-boxfor the repeated interaction by which collusion emerges. This is a modeling short-cut to justify also ourassumption that the side-contract is in fact enforceable even if there is no court of justice available to doso. This modeling characterizes the highest bound that can be achieved by the coalition.

7

Note �rst that we eliminate equilibria based on weakly dominated strategies at stage

3 of the overall game. Indeed, for any grand-mechanism G, there always exists a con-

tinuation equilibrium of the game of coalition formation in which each agent refuses any

collusive o�er he may receive because he expects that the other agent also refuses this

o�er anyway. Second, following Ai's rejection, Aj (j 6= i) may have updated his beliefs on

Ai's type. These beliefs a�ect the non-cooperative play of the grand-mechanism G and

therefore the status quo payo�s that Ai gets following a rejection of the side-mechanism

S. Therefore, there may exist several side-mechanisms o�ered as continuation equilibria

of the game of coalition formation depending on what is learnt following the rejection of

these side-mechanisms.

Let f~p1; ~p2g be a belief system where ~pi are agent A�i's beliefs on agent Ai if A�i con-

templates Ai's refusal to play the side-mechanism S. We denote by �(G; ~pi; p�i) the game

of asymmetric information induced by the grand-mechanism G at stage 4 following Ai's

refusal of playing S. In particular �(G; p; p) denotes this game of asymmetric information

when it is played with passive prior beliefs. E(G; ~pi; p�i) denotes the set of Bayesian-Nash

equilibria of �(G; ~pi; p�i). Finally, let us denote by Ui(�i; ei) the payo� of a �i agent Ai in

an equilibrium ei 2 E(G; ~pi; p�i). Note that this interim payo� is computed as an expec-

tation with respect to prior beliefs. Indeed, because joint deviations have probability zero

in a Bayesian-Nash equilibrium, nothing has been learned on A�i following Ai's refusal.

Therefore, the deviant agent Ai still continues to play the grand-mechanism with his prior

beliefs on A�i's type.

We are interested in collusive continuation equilibria in which no learning occurs from

the agreement of playing the side-mechanismS. We can thus de�ne similarly �(G�S; p; p)

the game of asymmetric information induced by the composition of the grand-mechanism

G and the side-mechanism S at stage 4 following acceptance of playing S by both agents.

E(G �S; p; p) denotes similarly the set of Bayesian-Nash equilibria of �(G �S; p; p). Since

the revelation principle applies at the last stage of the game, there is no loss of generality

in considering that E(G � S; p; p) contains the truthful equilibrium e�. Let Ui(�i) denote

agent Ai's payo� in this equilibrium when his type is �i.

3 No-Collusion and the First-Best Outcome

It is by now a well known result that the optimal mechanism achieves the �rst-best out-

come in this correlated environment when the implementation concept is Bayesian-Nash

equilibrium, even with interim individual rationality constraints (Crémer and McLean

(1988)).14 The key is to use agent A2's report, which is truthful in equilibrium, as a signal

correlated with agent A1's type and to condition A1's taxes on this information. The

14See also McAfee and Reny (1991) in the case of a continuum of types and Riordan and Sappington(1988) for a related model using ex post information in the case of only one agent.

8

�exibility in the taxes paid in the di�erent states of nature can then be used to �stochas-

tically� and costlessly deter any incentive to lie. If agent A1 does not report truthfully

his type, the mechanism is designed so that he gets a negative expected payo�. Since this

ingenious trick can be used for each agent simultaneously, the revelation of both agents'

types obtains at no cost for the principal.

Obviously, the taxes paid by the agents may be quite large when types become almost

uncorrelated.15 Indeed, the conditional probability that agents have the same types di-

minishes and the punishments (or rewards) in the di�erent states of nature must increase

to induce revelation.

We provide thereafter a simple proof of this �rst best implementation in our public

good setting. To do that, we �rst need to describe the set of implementable allocations in

this non-cooperative context. Without collusion, the revelation principle yields a charac-

terization of the set of implementable allocations with individual incentive compatibility

constraints only. Consider the Bayesian incentive compatibility constraint of agent A1

when he has a high valuation �� for the public good.16 Multiplying it by p(��) = p11+p12 > 0,

this constraint writes as:

p11(�t11 + ���x) + p12(�t12+ ��x) � p11(�t21 + ��x) + p12(�t22 + ��x); (1)

Similarly, when A1 has a low valuation � for the public good, his Bayesian incentive

compatibility constraint writes as (again multiplying by p(�) = p12 + p22 > 0):

p12(�t21 + �x) + p22(�t22+ �x) � p12(�t11 + ��x) + p22(�t12 + �x); (2)

Moreover, A1 must be induced to participate to the mechanism without knowing A2's

type. The following interim participation constraints must also be satis�ed: For a �� agent

p11(�t11 + ���x) + p12(�t12 + ��x) � 0; (3)

and for a � agent

p12(�t21 + �x) + p22(�t22 + �x) � 0: (4)

The principal maximizes expected welfare de�ned as:

SW = p11�2���x�2t11+(1+�)(2t11�c(�x))

�+2p12

�(��+�)x�t12�t21+(1+�)(t12+t21�c(x))

�

+p22�2�x� 2t22 + (1 + �)(2t22 � c(x))

�subject to constraints (1) to (4).17

Proposition 1 : Assume that types are strictly positively correlated, � > 0, then the

optimal provision of public good without collusive behavior entails:

� The �rst-best decision rule (�x�; x�; x�) where c0(�x�) = 2��, c0(x�) = �+ �� and c0(x�) = 2�.

� Participation constraints (3) and (4) are binding and both agent's types get zero rent.

15See the Appendix for explicit formulas of these taxes.16By symmetry A2 faces the same incentive and participation constraints.

9

Proof: All proofs are in an Appendix.

To implement the �rst-best outcome, the participation constraints (3) and (4) must be

binding and we look for solutions such that the incentive constraints (1) and (2) are also

binding.18 Then the existence of some strictly positive correlation imposes that the system

of linear binding constraints (1)-to-(4) is in fact invertible. This invertibility ensures that

the �rst-best schedule of public goods can be implemented at zero cost for the principal

The taxes which implement this optimal allocation of public good are highly dependent

on the information structure. When correlation becomes weaker, taxes become increas-

ingly punishing when both agents claim �� and when the agent who pays the tax claims

� and the other claims ��. On the contrary, agent Ai is increasingly rewarded when both

agents claim being � and when he claims �� and Aj (j 6= i) claims �. Indeed, with positive

correlation, it becomes easier to induce revelation from a �� agent Ai if, when he lies, he

is heavily punished when facing a �� agent Aj (j 6= i) who truthfully reveals.

Even a very small amount of correlation can be used to threaten the agents of being

heavily punished for lying on their types and to achieve the �rst best allocation. However,

with no correlation, allocative distortions become necessary to reduce a �� agent's costly

informational rent. Therefore, in this non-cooperative setting, the optimal mechanism

fails to be continuous with respect to the information structure.

For notational convenience, let p denote the probability of a high valuation type in

the case of no correlation. We have thus p11 = p2, p22 = (1� p)2 and p12 = p21 = p(1� p).

Proposition 2 : Assume that types are independently distributed, � = 0, then the optimal

provision of public good without collusive behavior entails:

� The second-best decision rule (�x�0; x�0; x

�0) where c

0(�x�0) = 2��, c0(x�0) = � + �� � �1+�

p

1�p��

and c0(x�0) = 2� � 2 �1+�

p

1�p��:

� A �� (resp. �) agent gets a strictly positive (resp. zero) rent.

Without any correlation, the system of binding equations (1) to (4) can no more be

inverted. Only expected taxes are de�ned from the binding incentive (1) and participation

constraints (4) and a �� agent's informational rent becomes costly. Allocative distortions

of x and x are needed to reduce this cost.

4 Collusion

Because the agents get zero rent from the mechanism proposed by the principal if they

play non-cooperatively, they are willing to coordinate their messages to countervail the

principal's power. The optimal grand-mechanism with a non-cooperative behavior creates

endogenously the stakes for some collusive behavior.

18Crémer and McLean (1988) show in fact that incentive constraints can be slack.

10

4.1 The Third-Party's Problem

Following Cramton and Palfrey (1995), we say that a side-mechanism S is unanimously

rati�ed for (e1; e2; ~p1; ~p2) if, for all �i 2 � and all i,

Ui(�i) � Ui(�i; ei)

where ei 2 E(G; ~pi; p�i) is a non-cooperative equilibrium of �(G; ~pi; p�i) and where

Ui(�i) =X��i

p(��ij�i)�yi(�i; ��i)� ti(�(�i; ��i)) + �ix(�(�i; ��i))

�;8�i 2 �;

is a �i agent Ai's payo� from playing e� 2 �(G � S; p; p).

De�nition 1 : A continuation collusive equilibrium of the game of coalition formation

consists of �rst, a system of beliefs f~p1; ~p2g and associated equilibria ei 2 �(G; ~pi; p�i)

(i 2 f1; 2g) and second, a truthtelling direct side-mechanism S� which is unanimously

rati�ed for the quadruplet (e1; e2; ~p1; ~p2) and which maximizes the third party's objective

function.

The continuation equilibrium of the game of coalition formation is thus a solution to

the third-party's following problem (denoted thereafter (T )):

maxf�(�);yk(�)(k2f1;2g)gX

(i;j)2f1;2g2

pij�� t1(�(�i; �j))� t2(�(�i; �j)) + (�i + �j)x(�(�i; �j))

�

subject to

(BIC) Ui(�i) �X��i

p(��ij�i)�yi(�i; ��i)� ti(�(�i; ��i)) + �ix(�(�i; ��i))

�;8(�i; �i) 2 �2;

(BIR) Ui(�i) � Ui(�i; ei) for some ei 2 E(G; ~pi; p�i) 8�i 2 �;

2Xk=1

yk(�i; ��i) = 0; 8(�i; ��i) 2 �2:

Along an equilibrium path on which the side-contract S� is unanimously rati�ed, no

learning occurs since unanimous rati�cation is a pooling strategy. Since nothing is revealed

at the rati�cation stage, Bayesian incentive constraints and �nal expected utilities are

computed with passive beliefs.

4.2 Weakly Collusion-Proof Mechanisms

De�nition 2 : G is weakly collusion-proof if and only if it is a truthtelling direct mecha-

nism and the null side-mechanism S�0 = f�� = Id; (y�k = 0)k2f1;2gg is unanimously rati�ed

for (e�; e�; p; p), where e� is the truthful equilibrium of G played with passive beliefs.

11

In other words, G is weakly collusion-proof if and only if the third-party o�ers the null

side-mechanism and there exists a perfect Bayesian equilibrium in E(G;S�0) such that

agents accept to play e� sustained with passive beliefs out of the equilibrium path.

Proposition 3 : To characterize the outcome of any perfect Bayesian equilibrium of the

game of grand-mechanism o�er cum coalition formation such that a collusive continuation

equilibrium occurs on the equilibrium path, there is no loss of generality in restricting the

principal to o�er weakly collusion-proof mechanisms.

The logic behind this weak collusion-proofness principle is similar to that underlying

the standard revelation principle: any equilibrium of the overall game of grand-mechanism

o�er cum side-contracting gives an allocation which can be replicated with a direct grand-

mechanism G o�ered by the principal himself. This grand-mechanism is such that the

coalition still forms at the rati�cation stage of �(G;S�0) and each agent Ai �nds optimal

to report his valuation �i truthfully to the principal thereafter.

One could argue that agents' information is not only restricted to their own valuations

but also includes the knowledge of the side-mechanism they use to collude. The principal

could try to elicit this information by also asking the agents which side-mechanism they

are actually playing. But the third-party could react by inducing further manipulations of

those reports of the side-mechanism. These reactions and counterreactions lead naturally

to a problem of in�nite regress. By restricting the principal to use grand-mechanisms only

contingent on the agents' valuations, we cut arbitrarily this process in favor of the collud-

ing agents. This amounts to an incomplete contracting assumption which �ts our desire to

give to collusive behavior its best chance.19 Within this incompleteness contractual frame-

work, we are nevertheless able to generalize the revelation principle and bene�t from the

weak collusion-proofness principle to obtain a simple constructive characterization of the

set of implementable allocations.

The next proposition characterizes this set in the case of symmetric grand-mechanisms.

Proposition 4 : A symmetric grand-mechanism G is weakly collusion-proof if and only

if there exists � 2 [0; 1[ such that:

�2t11 + 2���x � �t1(~�1; ~�2)� t2(~�1; ~�2) + 2��x(~�1; ~�2) 8(~�1; ~�2) 2 �2; (5)

�t12�t21+���+��

p11

p12���

�x � �t1(~�1; ~�2)�t2(~�1; ~�2)+

���+��

p11

p12���

�x(~�1; ~�2) 8(~�1; ~�2) 2 �2;

(6)

�2t22 + 2�� �

�p212p12p22 + �(p11p22 � p212)

���x

19In a related multiprincipal context, Epstein and Peters (1997) show that a similar in�nite regressmay converge. This convergence would allow one to de�ne universal sets of types and universal completegrand-mechanisms.

12

� �t1(~�1; ~�2)�t2(~�1; ~�2)+2���

�p212p12p22 + �(p11p22 � p212)

���x(~�1; ~�2) 8(~�1; ~�2) 2 �2; (7)

If � > 0, the Bayesian incentive compatibility constraint (1) of a �� agent is binding.

The weakly collusion-proof mechanisms described above are such that a � agent's

incentive constraint is not binding. Only a �� agent's incentive constraint may be binding.

These mechanisms are the only ones which are of interest as Proposition 5 will con�rm.

The parameter � is a discount factor less than one which captures the fact that col-

lusion takes place under asymmetric information. True valuations must be replaced by

virtual valuations20 in the coalition incentive constraints (6) and (7). The third-party

problem (T ) is constrained by the reservation utilities that the agents obtain from play-

ing non-cooperatively the grand-mechanism and the incentive compatibility constraint at

the coalition formation stage. Virtual valuations are then lower than true valuations to

take into account the costly multipliers of these constraints.

In the sequel, the downward coalition incentive constraints (rewritten after having

used the symmetry of the grand-mechanism) are of particular interest:

�2t11 + 2���x � �t12 � t21 + 2��x; (8)

�t12 � t21 +��� + � �

p11

p12���

�x � �2t22 +

��� + � �

p11

p12���

�x: (9)

(8) says that a coalition made with two �� agents prefers telling collectively the truth to

the principal rather than lying and telling that one of the agents is �. (9) says that a

(��; �) coalition prefers telling the truth rather than claiming that both agents are �.

The logic of the �rst-best non-cooperative implementation described in Section 2 is

to o�er large penalties and large rewards depending on the states of nature to induce

revelation. For instance, the formula for t11 (resp. t22) in the Appendix shows that this

tax may become extremely large and positive (resp. negative) when the agents' types are

less and less correlated. This suggests that the coalition incentive constraint (8) (resp.

(9)) is likely to be binding at the optimum of the principal's problem. The extent to

which the principal is restricted in using Bayesian transfers comes from the existence of

these coalition incentive constraints.

Once coalition incentive constraints are characterized, it is useful to derive necessary

monotonicity conditions for the implementability of a schedule of outputs.

Corollary 1 : For a weak correlation, � � (p12 + p22)p212p11

, the schedule of implementable

outputs is increasing (x � x � �x) for all � 2 [0; 1[. For a strong correlation, � >

(p12 + p22)p212p11

, the schedule of implementable outputs is non-monotonic (�x � x � x) if

and only if (�) = 1 + 2�p212

p12p22+�(p11p22�p212)� p11

p12� < 0, i.e., for � large enough, otherwise it

remains increasing.

20See Myerson (1979) for the standard de�nition of virtual valuations.

13

The striking feature of this corollary is that, in the case of strong correlation (i.e.,

when p12 is small enough), non-monotonic schedules of outputs can be implemented by the

principal if he chooses � large enough. The reason for this non-monotonicity is that virtual

valuations coming from the formation of the coalition under asymmetric information are

no longer ranked in the same order as true valuations. When agents' types are almost

perfectly correlated, the probability that they both get a high-valuation for the public

good is large. Henceforth, the incentive constraint of a �� agent willing to mimic a � one at

the coalition formation stage is also very costly to the third-party from an ex ante point

of view. Inducing revelation within the coalition requires therefore large distortions of

the optimal manipulation of reports ��(�; ��) with respect to what the third-party could

implement under symmetric information. Hence, the sum of the virtual valuations of a

(�; ��) coalition may become smaller than that of a (�; �) one.

This non-monotonicity property will create a substantive di�erence between collusion

under symmetric and asymmetric information when the correlation is strong contrary to

the case of a weak correlation.

5 The Optimal Weakly Collusion-Proof Mechanism

We now turn to some normative analysis and optimize the principal's welfare subject to

Bayesian individual incentive, participation and coalition incentive constraints. In the

sequel, we focus only on the �� agent's Bayesian incentive constraint (1), the downward

coalition incentive constraints (8) and (9) and the � agent's participation constraint (5).21

Before writing the principal's problem, let us �rst introduce four new variables �u =

�t11 + ���x; u1 = �t12 + ��x; u2 = �t21 + �x; u = �t22 + �x: These are the ex post

rents obtained by both types of agent in each possible state of nature. Rearranging

expected social welfare, individual and coalitional incentive and participation constraints

as functions of these variables, the principal's problem (P ) rewrites as:

maxfx(�);ug

p11�(1 + �)(2���x� c(�x))� 2��u

�+ 2p12

�(1 + �)((�� + �)x� c(x))� �(u1 + u2)

�

+p22�(1 + �)(2�x� c(x))� 2�u

�;

subject to

(BIC) p11�u+ p12u1 � p11u2 + p12u+��(p11x+ p12x) (10)

(CIC)1 2�u � u1 + u2 +��x (11)

(CIC)2 u1 + u2 � 2u +��x+p11

p12���(x� x) (12)

(IR) p12u2 + p22u � 0: (13)

21We check ex post that all other constraints are indeed satis�ed.

14

� For a weak correlation, � � (p12 + p22)p212p11

, the monotonicity condition derived from the

coalition incentive constraints is

�x � x � x:

� For a strong correlation, � � (p12 + p22)p212p11

, the monotonicity condition becomes

�x � x � x if and only if (�) � 0 and �x � x � x otherwise:

In the sequel, we focus separately on these two polar cases of weak and strong correlations.

5.1 Weak Correlation

Solving the principal's problem with standard Lagrangean technics yields:

Proposition 5 : Assuming that 0 � � � (p12 + p22)p212p11

, the symmetric optimal weakly

collusion-proof mechanism G� entails:

� A strictly decreasing schedule of outputs �xc > xc > xc with �no distortion at the top"

�xc = �x� and downward distortions with respect to the no collusion outcome otherwise:

xc < x� and xc < x� where

c0(xc) = �� + � ��

1 + ���

p11

2p12

�1 +

p12

p12 + �

�(14)

c0(xc) = 2� ��

1 + ���

1

p22

�p11 + 2p12 �

p12p11

p12 + �

�: (15)

� The Bayesian incentive constraint of a �� agent (10) is always binding. The downward

coalition incentive constraints (11) and (12) are both strictly binding when � > 0. The

participation constraint of a � agent (13) is also binding. All remaining constraints are

strictly satis�ed. A �� agent gets a strictly positive informational rent.

Binding Constraints: The fact that both coalitions (��; ��) and (�; ��) are prevented

from misreporting limits the feasible transfers that could be used by the principal to

extract the agents' information. t11 cannot be made largely positive as it is in the no-

collusion outcome without violating the coalition incentive constraint (11). A (��; ��) coali-

tion would like to avoid bearing these detrimental punishments by mimicking a (��; �)

coalition. (11) must be binding at the optimum. Similarly, a (��; �) coalition would like

to mimic a (�; �) one to get the corresponding large rewards requested in the no-collusion

outcome since t22 is then large and negative. (12) must thus also be binding.

Because of these constraints on the set of transfers, a �� agent must be given a strictly

positive rent contrary to the case without collusion. Large rewards and punishments can

no longer be used by the principal without violating the coalition incentive constraints.

Coalition Incentive Constraints: For a weak correlation, � = 0 at the optimum.

Indeed, there is no gain in having � strictly positive since this would only increase the cost

15

of the coalition incentive constraint (12).22 Interestingly, the binding collusion-proofness

constraints take therefore the same form as if agents could credibly share their information

within the coalition. Everything happens as if asymmetric information does not really

undermine the ability of the group to form. However, the agents' participation constraints

being the interim ones, they di�er from the case of symmetric information within the

coalition.

Output Distortions: Distorting downward x and x below their �rst-best values

reduces the costs of the individual (10) and coalitional incentive (11) and (12) constraints.

The size of the public good must be reduced because of asymmetric information and

collusion. Note that this result contrasts with the usual free-rider phenomenon discussed

in the literature. As shown for instance in Mailath and Postlewaite (1990), the reduction

in the size of the public good is then due to the con�ict between individual incentive and

individual participation constraints. Here, distortions come from the con�ict between

coalitional incentive and individual participation constraints. It is because a coalition can

form that individual incentives become costly to provide and that allocative distortions

are needed in this correlated environment.

Role of the Correlation: This con�ict between coalition incentive and participation

constraints increases with the amount of positive correlation between the agents' types.

When types are more positively correlated, the distortion required on xc to reduce the

cost of the coalition incentive constraint (11) is larger since p12 is smaller (see (14)). The

distortion on �xc needed to reduce the cost of the coalition incentive constraint (12) is

instead rather small since p22 is now relatively large (see (15)).

Ex post Rents: Ex post rents in the optimal weakly collusion-proof mechanism

satisfy:

�u < u2 +��xc (16)

u1 > u+��xc (17)

u2 < 0 (18)

�u > u1 (19)

u > 0 (20)

u2 > �u����xc (21)

u > u1 ���xc: (22)

(16) and (17) indicate that the truthful strategy of a �� agent is Bayesian. Indeed (16)

shows that a �� agent A1 never wants to tell the truth if he is sure of facing a �� agent A2.

Instead, (17) shows that he always tells the truth if he is sure that A2 claims �. Contrary

22In other words, the Bayesian incentive constraint in the third-party problem (T ) is not tight at theoptimum. This does not mean that (10) is not costly for the principal since he has a di�erent objectivefunction than that of the third-party.

16

to what happens in the no-collusion outcome where in fact �u < u1, a �� agent is now

rewarded when he faces a �� agent A2 and punished when he faces a � agent A2. It is only

because these rewards cover in expectation the penalties that a �� agent A1 weakly prefers

to tell the truth to the principal.

Instead, (21) and (22) show that the optimal weakly collusion-proof contract is such

that the dominant strategy incentive constraints of a � agent are always strictly satis�ed.

This dominant strategy requirement for one type somewhat simpli�es the optimal mecha-

nism since this type's equilibrium strategy is not sensitive to the exact beliefs that agents

have at the time of playing the mechanism.

Nevertheless, the ex post rent of a � agent A1 depends also explicitly on A2's type.

For instance, just as in the no collusion outcome, two � agents are given strictly positive

ex post rents. These agents are still subsidized for the consumption of the public good.

On the contrary, a � agent A1 facing a �� agent A2 makes a negative ex post pro�t. The

tax he pays for the public good is very large and his �nal utility is negative. Intuitively,

by setting u2 negative, the principal insures that (11) is not very costly. Satisfying the �

agent's interim participation constraint requires then to set u strictly above zero.

5.2 The Polar Case of No Correlation

In the degenerate case of no correlation, a rather striking result obtains:

Proposition 6 : With no correlation between the agents' types (� = 0), the optimal

weakly collusion-proof grand-mechanism G� entails the same strictly decreasing schedule

of outputs as without collusion, �xc0 = �x�0, xc0 = x�0 and xc0 = x�0.

Only the individual incentive (10) and the participation constraints (13) are strictly

binding. The multipliers of the coalition incentive constraints (11) and (12) are zero

at the optimum of (P ). Non-anonymous transfers implement in a Bayesian and weakly

collusion-proof way the optimal second-best contract, i.e., the non-cooperative outcome

obtained when there is no correlation between the agents' types. Collusion has no impact

when agents do not know more information on each other than what is available to the

principal. Intuitively, everything happens as if the principal sells the mechanism to the

third-party so that the latter perfectly internalizes the social welfare objective.

Strikingly, the optimal policy moves continuously from the case with positive corre-

lation to the case without correlation when coalition incentive constraints are taken into

account. Allowing collusion restores continuity of the optimal contract with respect to the

information structure. For any degree of positive correlation, there are enough binding

constraints to pin down the values of the optimal transfers in all states of nature. When

the degree of correlation converges to zero, these transfers converge and the limiting trans-

fers implement the second-best outcome in a collusion-proof way. Not only the expected

17

values of these limiting transfers but also their precise values in all states of nature are

now perfectly determined even in the case of no correlation.23

5.3 The Polar Case of Almost Perfect Correlation

With an almost perfect correlation, (p12 close to zero), both agents have almost always

the same type.

As a benchmark, suppose �rst that the agents can credibly exchange information on

their types because, for instance, the third-party is endowed with a technology making

this information veri�able within the coalition. Assume also that agents agree to play

the grand-mechanism before they learn each other types so that their participation con-

straints remain unchanged with respect to 5.1. Collusion-proofness constraints must now

be written with true valuations instead of virtual ones. This has two implications: First,

the binding downward coalition-proofness constraints take the same form as (11) and (12)

when � = 0. Second, monotonicity of the schedule of outputs (�x � x � x) is now always

necessary for implementability.

If this monotonicity constraint would not be binding at the optimum, the optimal

levels of public good would still be given by (14) and (15). However, when p12 is small

enough, it is easy to check that the monotonicity constraint x � x would be violated.

Hence, we get:

Proposition 7 : With almost perfect correlation between the agents' types (p12 small

enough but positive) and symmetric information within the coalition, the optimal weakly

collusion-proof grand-mechanism G� entails partial pooling �xc = �x� > xc = xc = xcP with

xcP de�ned by:

c0(xcP ) = 2p22� + p12(� + ��)

2p12 + p22� 2

�

1 + ����p11 + p12

2p12 + p22

�: (23)

Taking into account collusion-proofness constraints under symmetric information in an

almost perfectly correlated environment undermines quite signi�cantly the achievement

of the e�cient outcome. The optimal allocation entails now lots of pooling. When the

agents' types are strongly correlated, we have seen in Section 2 that rewards and punish-

ments necessary to implement the �rst-best outcome are relatively small. However, when

agents collude under symmetric information, their collusion becomes now also harder to

prevent. As a result, the optimal collusion-proof mechanism becomes less responsive to

their messages. In this framework, the optimal contract looks like an incomplete contract

making only partial use of the information.24

23Coalition incentive constraints somehow compactify the set of feasible allocations just as risk-aversionor limited liability does (see Robert (1991)).

24A last interpretation is worth stressing. Assume that the agents' types are now perfectly correlated(p12 = 0). The optimal pooling allocation becomes c0(xcP ) = 2� � 2 �

1+�

p11p22

��: This is the optimal

18

Things are quite di�erent under asymmetric information within the coalition. For

a strong correlation, non-monotonic schedules of outputs can be implemented by the

principal if � is large enough. Because now x < x, the collusion-proofness constraint (12)

is relaxed when � is as large as possible. An upper bound of the principal's welfare obtains

therefore for � = 1.

Proposition 8 : With almost perfect correlation and asymmetric information within the

coalition, the optimal weakly collusion-proof grand-mechanism G� entails: �xc = �x� > xc >

xc = 0 with xc de�ned by:

c0(xc) = 2� ��

1 + �

2p12p22

��:

The expected rent of the �� agent is p11�u+p12u1p11+p12

= p12p11+p12

��xc: Moreover, u2 = u = 0,

u1 =�1 � p11

p12

���xc and �u = ��xc.

For a strong correlation, the principal can now o�er non-monotonic schedules of ouputs

in a collusion-proof way. In particular, cancelling the production of the public good when

agents' types are di�erent and still keeping a positive production when types are the same

becomes a valuable option. This strategy is not very costly from an ex ante allocative point

of view since p12 is small. However, it relaxes quite signi�cantly (12) when � is positive.

In the limiting case where p12 is almost zero, the R.H.S. of (12) is so largely negative that

this coalition incentive constraint does not matter any more for the principal. u1 can be

set at a very large negative value and still this constraint can be easily satis�ed. Choosing

such a punishment in case of di�erent reports also relaxes quite signi�cantly (11) and the

principal's problem is almost as without collusion.

Since collusion does not a�ect too much social welfare, the principal can set xc to a

level close to its �rst best value x�. The punishment for claiming to have di�erent types

are so e�ective that a �� agent's expected rent is now close to zero. Therefore, with almost

perfect correlation, the optimal collusion-proof contract achieves an ex ante social welfare

close to its full information value.

6 Robustness

We focus in this section on the case of a small correlation.

6.1 Multiplicity of Equilibria of G�

When played with passive beliefs, the optimal weakly collusion-proof mechanism G� has

several pure strategy equilibria, i.e., jE(G�; p; p)j > 1. The �rst one is the truthful sym-

distortion in a one principal-agent model where the principal is facing directly a third-party endowedwith a utility function being the sum of the agents' utility functions.

19

metric equilibrium e� that we have derived above. However, there exist also two asym-

metric non-truthful pure strategy equilibria. e�1 (resp. e�2) is one such equilibrium where

A1 (resp. A2) always claims to be a � agent and A2 (resp. A1) reveals truthfully his type.

Indeed, from (21) and (22), a � agent always claims truthfully his type. A �� agent

A2 anticipates that A1 always claims �. Then, from (17), he reports truthfully. Thus,

agent A2 always reveals his type. Anticipating that A2 always reveal his type, from the

binding constraint (10), a �� agent A1 is indi�erent between lying or not and lies in this

equilibrium.

In this asymmetric equilibrium, denoted thereafter e�1, A1 gets the same ex ante and

interim payo�s as in e�, U1(�1; e�1) = U(�1) for all �1. Instead, A2's interim payo�s di�er

from that in e�. Indeed, we have U2(��; e�1) = u1 < U(��) = p11�u+p12u1p11+p12

(from (19)) and

U2(�; e�1) = u > U(�) = 0 (from (20)). However, using the values of ex post rents given in

the Appendix, A2's ex ante payo� is the same as in e�.

In fact, in equilibrium e�1, a (��; �) coalition reports (�; �) when it is indi�erent between

collectively claiming (��; �) and (�; �) ((12) is indeed binding for the ex post rents de�ned

by G�). This indi�erence of the coalition is broken in favor of the principal so that, among

all payo� equivalent equilibria from the third-party's point of view, the agents play the

most preferred by the principal. The fact that E(G�; p; p) is not a singleton is not a big

problem since all these equilibria yield the same aggregate payo� to the agents.25

6.2 Strong Collusion-Proofness

The fact that E(G�; p; p) is not a singleton shows also that G� is never strongly collusion-

proof, i.e., E(G�; S�0) is not reduced to a singleton either. Indeed, there exist other equi-

libria of �(G�; S�0) than unanimous rati�cation and subsequent play of e�. To show this

result, it is enough to exhibit such an equilibrium of �(G�; S�0). Unanimous veto of S�

0

and subsequent play of the equilibrium e�1 is such an equilibrium of �(G�; S�0) sustained

with passive beliefs.

This observation suggests two things. First, the rati�cation stage enlarges the set of

equilibria of G�. Second, we should test the robustness of G� to credible equilibria of the

game of coalition formation. We now turn to this issue.

6.3 Strong Rati�ability

As we have seen above, the rati�cation stage of �(G�; S�0) adds in fact a preplay commu-

nication stage to G�. At this stage, agents are allowed to make binary preplay announce-

ments (�Veto" or �Accept") which may signal some information on their types. Cramton

25This contrasts with Ma, Moore and Turnbull (1988) where Pareto-dominant non-cooperative and non-truthful equilibria may be a threat to the principal and must therefore be eliminated by using indirectmessage games.

20

and Palfrey (1995) have analyzed similar mechanism design problems and have proposed

the notion of strong rati�ability of a mechanism against itself to test its robustness to such

a cheap-talk stage.26 To understand this notion, let us �rst de�ne credible veto beliefs:

De�nition 3 : A belief system f~p1; ~p2g on � is a credible veto system of the truthful

decision rule e� if, for each i, there exist a non-cooperative Bayesian equilibrium ei 2

E(G�; ~pi; p�i) (with ei 6= e�) and refusal probabilities vi(�i), 8�i 2 �;8i 2 f1; 2g which

together satisfy:

1. vi(�i) > 0 for some �i 2 �;

2. vi(�i) = 1 for all �i 2 � such that U(�i) < Ui(�i; ei)

3. vi(�i) = 0 for all �i 2 � such that: U(�i) > Ui(�i; ei)

4. ~pi satis�es Bayes' rule, given the prior distribution p and the refusal probabilities

vi(�):

~pi(�ij�j) =

8>>>>>><>>>>>>:

p(�ij�j)vi(�i)X�k2�i;v(�k)>0

p(�kj�j)vk(�k)for �i such that vi(�i) > 0

0 for �i such that vi(�i) = 0:

Credibility of beliefs requires that vetoing e� should be interpreted as coming from the

subset of types who are the most likely to bene�t from this deviation when a non-deviant

agent's beliefs put all weight on this particular subset. This de�nition captures the ratio-

nal expectation reasoning that the di�erent types of the deviant agent make when they

envision vetoing the decision rule e�. Those types who gain from vetoing e� e�ectively

deviate and refuse to play e� with some probability when the non-deviant agent interprets

these deviations in a rational way. The set of such types is called a credible veto set. Still

following Cramton and Palfrey (1995), let also de�ne:

De�nition 4 : G� is strongly rati�able if and only if there does not exist a credible veto

system or, if for all credible veto sets and all credible veto beliefs ~pi, there exists a non-

cooperative equilibrium ei 2 E(G�; ~pi; p�i) such that U(�i) = Ui(�i; ei) for all i and for

all �i belonging to the credible veto set.

Strong rati�ability captures the idea that no type of agent is willing to credibly veto

the play of the truthful equilibrium e�. If it is strongly rati�able,G� is robust to cheap-talk

equilibria such that (possibly strict) subsets of types may deviate in a credible way.

26See Matthews and Postlewaite (1989) and Palfrey and Srivastava (1991) for other analyses allowingmore general cheap-talk stages in mechanism design.

21

De�nition 3 is actually a step in the construction of perfect sequential equilibria made

by Grossman and Perry (1986). Therefore, G� is strongly rati�able if and only if all

perfect sequential equilibria of �(G�; S�0) are payo� equivalent to unanimous rati�cation

of e�.

Proposition 9 : The optimal weakly collusion-proof grand-mechanism G� is strongly

rati�able.

The proof consists in describing the equilibrium correspondence E(G�; ~p1; p2) when

~p1 varies. From now on, we abuse notations and denote by ~p1 (resp. p) the probability

(resp. the conditional prior probability) that A2 assigns to A1 being ��. Starting from

E(G�; p; p) = fe�; e�1; e�2g, we get E(G�; ~p1; p) = fe�2g for optimistic beliefs ~p1 > p and

E(G�; ~p1; p) = fe�; e�1g for pessimistic beliefs ~p1 < p.

The charaterization of this correspondence makes easier to isolate the veto sets which

can credibly refuse to play e�. A �� agent A1 cannot be a credible veto set since G�

has a unique equilibrium e�2 when A2 holds optimistic beliefs on A1 and this equilibrium

gives to a �� agent A1 strictly less utility than e�. On the contrary, a � agent A1 can

be a credible veto set since G� has e� and e�1 for equilibria when A2 holds pessimistic

beliefs on A1. However, by refusing to ratify e�, this credible veto set cannot obtain more

utility than following unanimous rati�cation of e� since, with pessimistic beliefs, we have

E(G�; ~p1; p) = fe�; e�1g and both equilibria yield payo� U1(�; e�1) = U(�) to a � agent A1

anyway.

7 Conclusion

When agents collude to in�uence collective decision on public goods and their valuations

for the public good are positively correlated, there exists a trade-o� between e�ciency

and rent extraction. The optimal weakly collusion-proof contract depends on the degree

of correlation.

This correlation is also a crucial determinant of the group's ability to collude. In

particular, a strong positive correlation allows the principal to use asymmetric information

within the coalition to undermine signi�cantly its countervailing power.

Adding coalition incentive constraints restores also continuity between the correlated

and the uncorrelated information environments. Collusion does not matter in an uncor-

related environment with risk-neutral agents and the non-cooperative outcome is imple-

mentable in a collusion-proof way.

The bene�t from focusing on a discrete two type modeling of asymmetric information

is twofold. First, it has �rst given us a tractable characterization of the set of collusion-

proofness constraints. Second, it is also a key simpli�cation to test the robustness of

the optimal mechanism to some form of preplay communication since it becomes then

22

possible to fully describe the equilibrium correspondence of the optimal weak collusion-

proof mechanism G� when beliefs change.

Many lessons of this paper are independent of the speci�cs of the model, like the princi-

pal's objective function or the preferences of agents within the coalition. Therefore, these

results would also go through in other environments like auctions, regulation of duopoly,

design of incentive schemes within the �rm and arbitration mechanisms. More generally,

our results suggest that collusion stakes always exist in those correlated environments

and that the e�ciency of yardstick mechanisms depend signi�cantly on the amount of

correlation.

23

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mimeo, University of Toronto.

Green, J. and J.J. La�ont, 1977, �Characterization of Satisfactory Mechanisms for

the Revelation of Preferences for Public Goods,� Econometrica, 45: 427-438.

Green, J. and J.J. La�ont, 1979, �On Coalition Incentive Compatibility,� Review of

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Econometrica, 65: 875-911.

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Maximizing Mechanisms,� in Aggregation and Revelation of Preferences, ed. J.J.

La�ont, North-Holland, Amsterdam.

La�ont, J.J. and E. Maskin, 1980, �A Di�erential Approach to Dominant Strategy

Mechanisms,� Econometrica, 48: 1507-1520.

La�ont, J.J. and E. Maskin, 1982, �The Theory of Incentives: An Overview,� in Ad-

vances in Economic Theory, ed. W. Hildenbrand, Cambridge University Press.

24

La�ont, J.J. and J. Tirole, 1986, �Using Cost to Regulate Firms,� Journal of Political

Economy, 95: 921-937.

Ledyard, J. and T. Palfrey, 1996, �Interim E�ciency in a Public Goods Problem,�

mimeo, Caltech.

Ma, A., J. Moore and S. Turnbull, 1988, �Stopping Agents from Cheating,� Journal

of Economic Theory, 46: 355-372.

Mailath, G. and A. Postlewaite, 1990, �Asymmetric Information Bargaining Prob-

lems with Many Agents,� Review of Economic Studies, 57: 351-368.

Maskin, E., 1979, �Implementation and Strong-Nash Equilibrium,� Chapter 23 in Ag-

gregation and Revelation of Preferences, ed. J.J. La�ont, North-Holland.

Matthews, S. and A. Postlewaite, 1989, �Pre-Play Communication in Two-Person

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Riordan, M. and D. Sappington, 1988, �Optimal Contracts with Ex Post Informa-

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bridge University Press.

25

Figure 1: Game Tree

(U1(�1; e�); U2(�2; e�))

(U1(�1; e2); U2(�2; e2))

(U1(�1; e1); U2(�2; e1))

(U1(�1; e0); U2(�2; e0))

�(G � S; p; p)

�(G; ~p1; p)

�(G; p; ~p2)

�(G; ~p1; ~p2)

������

������

������������HHHHHHHHHHHH

YN

YN

A2

�

�

�

�

Y N

�A1

?

S

T

���@@@ �

��@@@

�����@

@@@@�

���

YN NY

(0; 0) (0; 0) (0; 0)

A2

Y N

�A1

?

�

?

Nature (�1; �2)

G

P

llllll

llllll

9>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>;

�(G;S)

�

26

Appendix

Proof of Proposition 1: When (3) and (4) are binding the principal's expected welfare

rewrites as SW = (1 + �)�p11(2���x � c(�x)) + 2p12

�(�� + �)x� c(x)

�+ p22(2�x� c(x))

�if

there exist transfers such that the incentive compatibility constraints (1) and (2) hold.

Optimizing this expression yields the �rst-best decision rule. Since we look for transfers

such that (1) and (2) are also binding, (t11; t12; t21; t22) must solve:

p11t11 + p12t12 = ��(p11�x+ p12x); (24)

p11t21 + p12t22 = ��(p11x+ p12x); (25)

p12t21 + p22t22 = �(p12x+ p22x); (26)

p12t11 + p22t12 = �(p12�x+ p22x): (27)

Those equalities are satis�ed for some transfers when the matrix

p11 p12p12 p22

!

is invertible. This holds since its determinant is � = p11p22�p212 > 0. Solving for the taxes:

t11 =1�

���(p11�x+p12x)p22��(p12�x+p22x)p12

�; t12 =

1�

��(p12�x+p22x)p11���(p11�x+p12x)p12

�;

t21 = 1�

���(p11x + p12x)p22 � �(p12x + p22x)p12

�; t22 = 1

�

��(p12x + p22x)p11 � ��(p11x +

p12x)p12�: These taxes become very large as � goes to zero. Indeed, we have �t11+ ���x =

�1���p12(p12�x + p22x); and �t22 + �x = 1

���p12(p11x + p12x): Hence, t11 goes towards

plus in�nity and t22 goes towards minus in�nity when � goes to zero. Similarly, t21 goes

towards plus in�nity and t12 goes towards minus in�nity when � goes to zero.

Proof of Proposition 2: The proof is standard and thus omitted.

Proof of Proposition 3: Let us consider a perfect Bayesian equilibrium of the

overall game of grand-mechanism o�er cum coalition formation such that a side-contract

is unanimously rati�ed. This P.B.E is in fact a triplet fG�;S�; (~p1; ~p2)g where:

� G� fromM =M1�M2 into D = X�T 2 maps the messages (m1;m2) sent by the agents

into an allocation (public good, taxes). G� maximizes the principal's welfare taking into

account the continuation equilibrium of the game of coalition formation.

� S� is a side-mechanism which, since the revelation principle applies at the last stage

of the game, can be taken as being a direct mechanism mapping � � � into the set of

measures on message spaces. Let e� be its truthful equilibrium. S� maximizes the sum of

the agents' expected utilities subject to individual Bayesian incentive constraints, budget

balance and individual rationality constraints Ui(�i) � Ui(�i; ei) for some ei 2 E(G; ~pi; p�i)

and some f~p1; ~p2g:

� f~p1; ~p2g is the system of out-of equilibrium posterior beliefs used respectively by agent

A2 and A1 to assess respectively A1 and A2's type following their respective veto of the

27

side-mechanismS�. These are the beliefs used in the non-cooperative play ofG to compute

the status quo payo�s Ui(�i; ei).

� Consider now the new grand-mechanism ~G = G� � S�. We prove that there exists a

P.B.E. of the overall game of contract o�er cum coalition formation in which the principal

o�ers ~G which is a direct mechanism from ��� into D = X �T 2, the third-party o�ers

the null side-mechanism S�0 = f� = Id; yk = 0g and this choice is sustained by passive

beliefs, ~p1 = ~p2 = p. Because S� solves (T ) with reservation utilities Ui(�i; ei), the null

side-mechanism solves (T ) with reservation utilities Ui(�i). Indeed, suppose it is not the

case, then there would exist a side-mechanism ~S such that the third-party can achieve a

strictly greater payo� for the coalition than with S�. Since by de�nition Ui(�i) � Ui(�i; ei)

the third-party's payo� from o�ering S� � ~S in the �rst place would be strictly greater

than that achieved with S�. This would contradict that S� is optimal when G� is o�ered.

� Hence, o�ering the grand-mechanism ~G insures to the principal that there is a P.B.E.

of the continuation game sustained with passive beliefs in which the null-side-mechanism

is unanimously rati�ed.

Proof of Proposition 4 and Corollary 1: We �rst solve for the third-party's

optimal side-mechanism sustained by a reversion to a non-cooperative equilibrium of

the mechanism G(�) played with passive beliefs. Then, we identify conditions such that

S�0 = f� = Id; yk = 0g, i.e., the null side-mechanism, is the solution to (T ). We conclude

by deriving the monotonicity conditions that must be satis�ed by such a weakly collusion-

proof contract.

� Since, we are not interested in grand-mechanisms such that the � agent incentive con-

straint is binding (this constraint will be satis�ed ex post), we write the third-party's

problem as (the lowerscript denotes the index of the agent concerned with the transfer

and let �(�i; �j) = �ij for simplicity):

maxf�(�);y(�)g

X(i;j)2f1;2g2

pij�� t1(�ij)� t2(�ij) + (�i + �j)x(�ij)

�

subject to

� Budget-balance:2X

k=1

yk(�i; �j) = 0 8(�i; �j) 2 �2; (28)

� Incentive constraints for respectively the �� agents A1 and A2:

p11�� t1(�11)� y1(��; ��) + ��x(�11)

�+ p12

�� t1(�12)� y1(��; �) + ��x(�12)

�

� p11�� t1(�21)� y1(�; ��) + ��x(�21)

�+ p12

�� t1(�22)� y1(�; �) + ��x(�22)

�; (29)

p11�� t2(�11)� y2(��; ��) + ��x(�11)

�+ p21

�� t2(�21)� y2(�; ��) + ��x(�21)

�� p11

�� t2(�12)� y2(��; �) + ��x(�12)

�+ p21

�� t2(�22)� y2(�; �) + ��x(�22)

�; (30)

28

� Participation constraints for respectively the �� agents A1 and A2:

p11��t1(�11)�y1(��; ��)+��x(�11)

�+p12

��t1(�12)�y1(��; �)+��x(�12)

�� (p11+p12)U1(��; e1);

(31)

p11��t2(�11)�y2(��; ��)+��x(�11)

�+p21

��t2(�21)�y2(�; ��)+��x(�21)

�� (p11+p21)U2(��; e2);

(32)

for some equilibria ei 2 �(G; p; p) (i 2 f1; 2g).

� Participation constraints for respectively the � agents A1 and A2:

p21��t1(�21)�y1(�; ��)+�x(�21)

�+p22

��t1(�22)�y1(�; �)+�x(�22)

�� (p21+p22)U1(�; e1);

(33)

p12��t2(�12)�y2(��; �)+�x(�12)

�+p22

��t2(�22)�y2(�; �)+�x(�22)

�� (p12+p22)U2(�; e2):

(34)

Let introduce the following multipliers � (�i; �j) for (28), �i for (29) and (30). ��i for (31)

and (32) �i for (33) and (34). We write the Lagrangean L of the maximization problem

above as:

L = E(U1+U2)+2X

i=1

�i (BIC)i(��)+

2Xi=1

��i (BIR)i(��)+

2Xi=1

�i (BIR)i(�)+X

(�i;�j)

� (�i; �j) (BB) (�i; �j):

Optimizing with respect to y1(��; ��) and y2(��; ��) yields respectively:

� (��; ��)� p11(�1 + ��1) = 0; (35)

� (��; ��)� p11(�2 + ��2) = 0: (36)

Optimizing with respect to y1(��; �) and y2(��; �) yields respectively:

� (��; �)� p12(�1 + ��1) = 0; (37)

� (��; �) + p11�2 � p12�2 = 0: (38)

Optimizing with respect to y1(�; ��) and y2(�; ��) yields respectively:

� (�; ��) + p11�1 � p21�1 = 0; (39)

� (�; ��) + p21(�2 + ��2) = 0: (40)

Finally, optimizing with respect to y1(�; �) and y2(�; �) yields respectively:

� (�; �) + p12�1 � p22�1 = 0; (41)

� (�; �) + p21�2 � p22�2 = 0: (42)

� Optimizing with respect to �11 yields:

��11 2 argmax~�11

p11�� t1(~�11)� t2(~�11) + 2��x(~�11)

�

29

+(��1 + �1)p11�� t1(~�11) + ��x(~�11)

�+ (��2 + �2)p11

�� t2(~�11) + ��x(~�11)

�:

Taking into account (35) and (36), �1 + ��1 = �2 + ��2, and simplifying yields:

��11 2 argmax~�11

��t1(~�11)� t2(~�11) + 2��x(~�11)

�: (43)

� Optimizing with respect to �12 yields:

��12 2 argmax~�12

p12�� t1(~�12) � t2(~�12) + (�� + �)x(~�12)

�

+p12(�1 + ��1)�� t1(~�12) + ��x(~�12)

�� �2p11

�� t2(~�12) + ��x(~�12)

�+�2p12

�� t2(~�12) + �x(~�12)

�: (44)

Using (37) and (38), �1 + ��1 = ��2p11p12

+ �2: Inserting into (44) yields:

��12 2 argmax~�12

�t1(~�12)� t2(~�12) +

��� + � �

p11�1

p12���x(~�12)

!(45)

with �1 =�2

1+�1+��1: Similarly

��21 2 argmax~�21

�t1(~�21)� t2(~�21) +

��� + � �

p11�2

p21���x(~�21)

!(46)

with �2 =�1

1+�2+��2:

� Optimizing with respect to �22 yields:

��22 2 arg max~�22

p22�� t1(~�22)� t2(~�22) + 2�x(~�22)

�+ p22�1

�� t1(~�22) + �x(~�22)

�

+p22�2��t2(~�22)+�x(~�22)

��p12�1

��t1(~�22)+��x(~�22)

��p21�2

��t2(~�22)+��x(~�22)

�: (47)

Note again that �1 + ��1 = ��2p11p12

+ �2 and �2 + ��2 = ��1p11p12

+ �1. Using (41) and

(42), one also gets p22(1 + �1) � p12�1 = p22(1 + �2) � p12�2 = p22�1 + �1 + ��1

�+ �2�

p12=

p22�1 + �2 + ��2

�+ �1�

p12= B. Simplifying into (47) yields then:

��22 2 argmax~�22

�t1(~�22)� t2(~�22) + 2�x(~�22)�

(�1 + �2)p12B

��x(~�22)

!:

Put di�erently,

��22 2 arg max~�22

0@�t1(~�22)� t2(~�22) +

0@2� � �1p12

p22 + �1�

p12

�� ��2p12

p22 + �2�

p12

��

1Ax(~�22)

1A :(48)

In the sequel, we consider symmetric grand-mechanisms such that �1 = �2 = � and

t1(�1; �2) = t2(�2; �1).27 We then have:

��22 2 argmax~�22

0@�t1(~�22)� t2(~�22) + 2

0@� � �p12

p22 + � �

p12

��

1A x(~�22)

1A : (49)

27Asymmetric mechanisms are discussed in the Proof of Proposition 5.

30

� In a weakly collusion-proof mechanism ��ij = (�i; �j). Inserting into (43), (45), (46) and

(49) yields constraints (5), (6) and (7).

� Note that � = �1+�+�� 2 [0; 1[: Moreover, � > 0 when the Bayesian incentive constraints

(29) and (30) are binding in (T ).

� Note also that (31), (32), (33) and (34) are binding for a weakly collusion-proof mech-

anism. Hence, for such a mechanism, the slackness conditions obtained from the La-

grangean's optimization do not give any information on �. Therefore, the principal has

some �exibility in choosing this variable.

� We now check for the monotonicity of the schedule of outputs when the mechanism is

weakly collusion-proof. From (43) and (44) taken for ��ij = (�i; �j), we have:

�2t11 + 2���x � �t12� t21 + 2��x

�t12 � t21 +��� + � �

p11

p12���

�x � �2t11 +

��� + � �

p11

p12���

��x:

Summing these two inequalities yields���1+ p11

p12�

�(�x�x) � 0; which is satis�ed for �x � x

(since � � 0). Proceeding in a similar way and using (44) and (49) for the coalitions (��; �)

and (�; �), another revealed preference argument tells us that�1 + 2p212�

p12p22+�(p11p22�p212)� p11�

p12

�(x�x) � 0: Denoting by (�) the �rst term on the left-hand-side of the latter inequality.

(�) is concave in � (since 00(�) =�4p312p22�

(p12p22+��)3< 0). Hence, it is either minimum at 0 or

1. (0) = 1 > 0 and (1) > 0 if and only if � < (p22 + p12)p212p11

. For a weak correlation,

namely � < (p22+p12)p212p11

, (�) is always positive and the monotonicity condition becomes

x � x. For a strong correlation, namely � > (p22 + p12)p212p11

, (�) is negative for � close

enough to one and the monotonicity condition becomes then x � x.

Proof of Proposition 5:

� We denote by �; �; ; �, the multipliers respectively of (10), (11), (12) and (13).

Optimizing with respect to �u yields:

2�p11 = �p11 + 2�: (50)

Optimizing with respect to u1 yields:

2�p12 = �p12 � � + : (51)

Optimizing with respect to u2 yields:

2�p12 = ��p11 � � + + �p12: (52)

Optimizing with respect to u yields:

2�p22 = ��p12 � 2 + �p22: (53)

31

Summing (50) to (53) yields:

� =2�

p12 + p22> 0: (54)

Inserting this latter expression into (52) yields:

2�p12

�1�

1

p12 + p22

�= ��p11 � � + : (55)

Subtracting (51) from (55) yields:

� =2�p12

(p12 + p22)(p11 + p12)> 0: (56)

Inserting this expression into (50) yields then:

� = �p11

�1 �

p12

(p12 + p22)(p12 + p11)

�> 0: (57)

� and � are strictly positive when � = (p12 + p22)(p12 + p11)� p12 > 0: Finally, inserting

(57) into (51) gives:

= �(p11 + 2p12)�1 �

p12

(p12 + p22)(p12 + p11)

�> 0: (58)

� Since > 0 and since the monotonicity condition implies that x � x � 0 for a weak

correlation, � = 0 minimizes the cost of constraint (12).

� Optimizing with respect to �x; x and x yields c0(�x) = 2��; c0(x) = ��+ �� 11+�

12p12

(�p11+

�)��; and c0(x) = 2� � 11+�

1p22

(�p12 + )��: Using (56), (57) and (58) yields (14) and

(15).

Monotonicity of outputs obtains when 1� �1+�

p112p12

�1+ p12

p12+�

�> � �

1+�1p22

�p11+2p12�

p12p11p12+�

�: Because � � 0, this property holds when 1 > �

1+�

�p11p12

� 2p12p22

�. Because � � 0, the

latter inequality also holds when 1 > p11p12

� 2p12p22

or when � < p12(p12 + p22). However, for

a weak correlation, we have by de�nition � �p212p11

(p12+ p22). But this latter left-hand-side

is lower than p12(p12 + p22) when p12 � p11. Hence, monotonicity of outputs is ensured.

� Ex post rents are obtained from solving the system (10) to (13) (with � = 0). After

tedious computations, we �nd:

u2 = �p11p22

2(p12 + �)��(xc � xc) < 0: (59)

u =p11p12

2(p12 + �)��(xc � xc) > 0: (60)

u1 � u���xc = u� u2 =p11

2(p11 + p12)��(xc � xc) > 0: (61)

�u� u2 ���xc = �p12

2(p11 + p12)��(xc � xc) < 0: (62)

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u2 � �u+���xc = ��(�xc � xc) +p12

2(p11 + p12)��(xc � xc) > 0: (63)

u� u1 +��xc =p11 + 2p122(p11 + p12)

��(xc � xc) > 0: (64)

� Using (63) and (64), it is immediate to show that a � agent's incentive constraint is

strictly satis�ed. A �� agent receives a strictly positive rent in this truthful equilibrium:

U(��) = 1p11+p12

(p11�u+p12u1) = ���p11x

c+p12xc

p11+p12

�� �p11

p12+���(xc�xc) which is strictly positive

since the factor of x is strictly positive and the factor of x is also strictly positive (p12 +

� � �(p11 + p12) > 0 when � > 0 and p11 + p12 < 1).

Lastly, monotonicity of the decision rule and the fact that (11) and (12) are binding

ensure that the other coalition incentive constraints are satis�ed.

Asymmetric Grand-Mechanisms: One may wonder whether the principal could not

prevent collusion in a cheaper way by o�ering a grand-mechanism which treats di�erently

both agents still keeping a symmetric decision rule. In particular, we would have some

�exibility in setting �1 6= �2 and asymmetric transfers in all states of nature. For a weak

correlation, one can show that the same monotonicity conditions as under symmetric

information within the coalition can be derived, namely �x � x � x. Again, the costs of the

coalition incentive constraints involving a (�; ��) coalition are minimized when �1 = �2 = 0.

There is then a simple argument showing that, in fact, there is always a symmetric

mechanism which does at least as well for the principal. First note that the principal's

payo� depends only on the sum of the agents' ex post rents. Suppose indeed that the

optimal weak collusion-proof grand-mechanism is then asymmetric. Let us denote by G�1

this mechanism. Because of symmetry, another mechanism G�2 obtained by permuting

all indices is also a solution. The grand-mechanism 12G

�1 +

12G

�2 obtained by averaging ex

post rents with an equal weight is symmetric. Moreover, it is also individually incentive

compatible for both agents and satisfy the same coalition incentive constraints as G�1.

Hence, it achieves the same welfare as G�1 and G

�2. Finally, there is no loss of generality

in looking for optimal symmetric mechanisms in the case of a weak correlation.

Proof of Proposition 7: Monotonicity of outputs obtains when 1 � �1+�

p112p12

�1 +

p12p12+�

�> � �

1+�1p22

�p11 + 2p12 �

p12p11p12+�

�: This condition does not hold when p12 is small

enough, i.e., for strong correlation. Pooling arises at the optimum. Solving for the solution

of the principal's problem under the cons]TJ aint that x = x = xP yields the �rs] order

condition: (2p12+p22)c0(xP ) = 2p22�+2p12(�+��)� 11+�

(�(p12+p11)+�+ )��: Simplifying,

we get (16).

Proof of Proposition 8: The optimal con]TJ act is obtained for a non-monotonic

schedule of outputs, �x > x > x. Then the principal �nds optimal to choose � = 1 to

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minimize the cost of (12).28 The binding constraints of (P ) are then:

(BIC) p11�u+ p12u1 � p11u2 + p12u+��(p11x+ p12x) (65)

(CIC)1 2�u � 2u+ 2��x (66)

(CIC)2 u1 + u2 � 2u +��x+p11

p12���(x� x) (67)

where � = 1

(IR) p12u2 + p22u � 0: (68)

� Let denote by respectively �, �, and � the multipliers of these constraints. Proceeding

as in Proof of Proposition 5, we �nd � = 2�p12(p11+p12)(p22+p12)

, � = �p11�

p12+�, = 2�p12�

p12+�,

� = 2�p12+p22

.

� Solving for the optimal outputs yields �xc = �x� and c0(xc) = �+��� �1+�

p11p12+�

�1 + 1

p12

���

when p12 is small enough, the positiveness constraint xc � 0 is binding and xc = 0 is in

fact optimal. Finally, we have also: c0(xc) = 2�� �1+�

2p12p22

��: Hence, xc converges towards

x� when p12 becomes arbitrarily small.

� Solving for the values of the ex post rents, u = u2 = 0, u1 = ��xc�1 � p11

p12

�and

�u = ��xc. It is routine to check that all neglected Bayesian incentive, participation and

collusion-proofness constraints are satis�ed.

Proof of Proposition 9: To derive the equilibrium correspondence of G� when

posterior beliefs change, �rst, note that for any belief system, a � agent (whether a deviant

or a non-deviant one) reports truthfully because of dominant strategy for this type. Hence,

non-truthful equilibria obtains when a �� agent A2 (the non-deviant agent) lies and claims

he is �. Since a �� agent A2 is indi�erent between claiming � and �� when G� is played

with passive beliefs ((10) is binding for G�) and since (16) holds, a �� agent A2 prefers to

claim � when G� is played with optimistic beliefs ~p1 > p. Then, since (17) and (22) hold,

A1 whatever his type reports truthfully. It is thus immediate that E(G�; ~p1; p) = fe�2g for

optimistic beliefs ~p1 > p. For pessimistic beliefs, a �� agent A2 always prefers to report

instead his true type ��. Non-truthful equilibria where A2 lies cannot hold. However, the

deviant agent A1 still having passive beliefs on A2 and being indi�erent between lying or

not (since (10) is binding in G�) may also lie without changing A2's incentives to tell the

truth. Thus, we have E(G�; ~p1; p) = fe�; e�1g for pessimistic beliefs.

28� belongs to the open interval [0; 1[ but can be made as close as possible to one by increasing themultiplier � of the incentive constraint in the third party's problem.

34